On September 7, 2018 I gave a talk in the CUNY Set Theory Seminar in New York.

*How to obtain Woodin cardinals from the determinacy of long games*

*Abstract:* We will study infinite two player games and the large
cardinal strength corresponding to their determinacy. For games of
length $\omega$ this is well understood and there is a tight
connection between the determinacy of projective games and the
existence of canonical inner models with Woodin cardinals. For games
of arbitrary countable length, Itay Neeman proved the determinacy of
analytic games of length $\omega \cdot \theta$ for countable $\theta
> \omega$ from a sharp for $\theta$ Woodin cardinals. We aim for a
converse at successor ordinals and sketch how to obtain $\omega+n$
Woodin cardinals from the determinacy of $\boldsymbol\Pi^1_{n+1}$
games of length $\omega^2$. Moreover, we outline how to generalize
this to construct a model with $\omega+\omega$ Woodin cardinals from
the determinacy games of length $\omega^2$ with
$\Game^{\mathbb{R}}\boldsymbol\Pi^1_1$ payoff.

This is joint work with Juan P. Aguilera.